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\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

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\Title Hyperbola.

\lf
See also Parabola, Ellipse, Conic Sections and their ATOs.

\lf
The most common parametric equations for a Hyperbola with semi-axes aa
and bb are:

\noindent
$x(t) =  \pm aa\cosh(t), \hskip0.25cm
 y(t) = bb \sinh(t) ,\hskip1cm  t\in \Bbb R$;

\noindent
and another version is: 

\noindent
$x(t) = aa/\cos(t),  \hskip0.1cm
y(t) = bb \sin(t)/\cos(t) ,\hskip0.2cm  t\in [0,2\pi] $.

\noindent
The corresponding implicit equation is:

\noindent
$ ({x\over aa})^2 - ({y\over bb})^2 = 1$.

\noindent
The function graphs: $\{(x,y);\ y = 1/x + m\cdot x \}$
are also Hyperbolae.

\noindent
A geometric definition of the Hyperbola is: 


{\narrower\noindent
A Hyperbola is the set of points for which the {\bf difference of the
distance} from two focal points is constant. \par\noindent}
\noindent
Or: 

{\narrower\noindent
A Hyperbola is the set of points which have the {\bf same distance} from
a circle and a (focal) point {\bf outside} that circle. \par\noindent}
\bigskip\bigskip \goodbreak

\noindent
If one applies an inversion $ (x,y) \to (x,y)/(x^2+y^2)$ to a right Hyperbola
(i.e. $aa=bb$) then one obtains a Lemniscate. 

 \lf
In the visualization of the complex map $z\to z+1/z$ in Polar
Coordinates, the image of the radial lines are the Hyperbolae:

\lf
$ x(R) = (R+1/R)\cos\phi  \lf
  y(R) = (R-1/R)\sin\phi,\hskip1cm R\in \Bbb R $.

  \lf
And the image of the standard Cartesian Grid under the complex map
$z\to \sqrt z$ is a grid of two families of orthogonal Hyperbolae.

\noindent
H.K.



\bye
